## Linear Operators: General theory |

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Page 193

12

ZT, X). Let E be a g-null set in R. Then for ).-almost all t, the set E(t) = t] e E) is a fi-

null set. Proof. By

12

**Lemma**. Let (R. ZR, g) be the product of finite measure spaces (5, E, fi) and (T,ZT, X). Let E be a g-null set in R. Then for ).-almost all t, the set E(t) = t] e E) is a fi-

null set. Proof. By

**Lemma**11 it may be assumed that the measure spaces are ...Page 697

11

be a strongly measurable semi-group of operators in ^(S, E, fi) with |7,(<1, . . ., <J|i

^ 1. . . ., ^ 1. Let 1 g p < oo, / e L„, and f*(s) = sup \A{a.){f, s)\ where 0<a<oo AM ...

11

**Lemma**. Let (S,Z,/x) be a positive measure space and let {r(<!, . . ., tk), tk > 0}be a strongly measurable semi-group of operators in ^(S, E, fi) with |7,(<1, . . ., <J|i

^ 1. . . ., ^ 1. Let 1 g p < oo, / e L„, and f*(s) = sup \A{a.){f, s)\ where 0<a<oo AM ...

Page 699

Therefore dx which proves (*) and completes the proof of the

shall now state and prove the

occurring later the following

Therefore dx which proves (*) and completes the proof of the

**lemma**. Q.E.D. Weshall now state and prove the

**lemma**referred to as CPk. For technical reasonsoccurring later the following

**lemma**is stated for what might be called a positive ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero